We generalize a recently investigated lattice model of polydisperse semiflexible linear polymers formed under equilibrium polymerization in a solution [Gujrati, P. D.; Rane, S. S.; Corsi, A. Phys. Rev. E 2003, 67, 052501] and conduct a comprehensive investigation of its melting properties. Each polymer has two end groups and at least one chemical bond. The model is characterized by six energies, three of which are for the interaction between the middle group, the end group, and the solvent, and the remaining three represent conformational energies for a gauche bond, a hairpin turn, and a pair of neighboring parallel bonds. Two activities control the end group and the middle group densities and give rise to polydisperse chains whose chain lengths and numbers are not fixed. We study the melting properties such as the melting temperature, the latent heat, and the energy and entropy of fusion at fixed pressure as a function of various model parameters such as the monomer interactions, nature of end groups, chain rigidity, solvent quality, and degree of polymerization. Our theory is thermodynamically consistent in the entire parameter space and improves upon the classical theories; hence, our results should prove useful.